 So, in the decomposition of the elliptic genus of K3, we encountered an object which I said was a Mach modular form, and I should define what that is. Now, there's a very interesting colorful history, which I'm not going to tell you about, but you can read about in various places. These objects were invented by Ramanujan in the last year of his life and described in a letter that he wrote to Hardy, but he didn't define them. He just wrote down examples, and it took mathematicians a long time to figure out what he was doing and a more rigorous mathematical framework for them, and I'll be telling you about the results of that investigation. Although, I've talked to mathematicians who claim that the correct definition is not yet understood, that there should be something better. But anyway, I can tell you what there is. So, let me just, I think earlier on, I should have defined a modular form. I kind of assumed it was something that you knew, but just to make sure that we both, that we're all on the same page, a modular form of what K is a holomorphic function from the upper half plane, the complex numbers, which obeys this modular transformation property, and we sometimes weaken this or vary this in various ways, so we sometimes talk about weekly holomorphic modular forms. That means that when we look at f tau, f of tau and write a Q expansion for it, we allow some negative powers of Q, so we might have a sum here that starts at n greater than some minus n. These are called weekly holomorphic. And the other thing that we can do here is we could insert some row here, which is either a phase, or if we want to look at, it could be a matrix, we're looking at vector-valued modular forms. A matrix meaning that we have some matrix that depends on an element of SL2z, gives us some representation of that, which mixes the components of f. So these are natural generalizations that occur in conformal field theory. All right, so what is a Mach modular form? Well, there are various ways to define it, but here's one way. You start with an object, which is kind of like a modular form, but different. You start with something that's called a weight k-harmonic mass form. I'm not sure if it's mass or mass, but anyway it's M-A-A-S-S, which is the name of a mathematician. And a weight k-harmonic mass form is a function, not necessarily holomorphic, from the upper half plane to the complex numbers, such that it transforms under modular transformations like a modular form, perhaps with some phase, some multiplier system, or you could consider vector-valued generalizations, but I won't worry about those subtleties right now. And it obeys the equation that the weight k-laplacian y to the 2 minus k d by d tau y to the k d by d tau bar is equal to 0 on F. So here the weight k-laplacian is just the laplacian acting on k-forms on the upper half plane. You write tau as x plus iy, then we use the usual metric that has constant negative curvature on the upper half plane, and delta k is the associated weight k-laplacian. So you could think of this as suppose you were doing quantum mechanics, but instead of doing quantum mechanics on the line or the plane with the flat metric, you were doing quantum mechanics on the upper half plane. If you're doing quantum mechanics on the upper half plane and you didn't have a potential, then the eigenfunctions would be solutions to the laplacian, and these would be like zero-energy states in quantum mechanics. So that's kind of, you know, you could think of them sort of like that. So the mass form means it's an eigenvalue of the weight k-laplacian. The weight k means it transforms like a weight k-modular form, and harmonic means that it has eigenvalue 0 under the weight k-laplacian. I'll let mk hat denote the space of weight k-harmonic mass forms. So now we also consider the space k with a little bang there, which is the space of weakly holomorphic weight k-modular forms. All right, so now what we're going to do is, given one of these weight k-harmonic mass forms, I haven't told you how to find them yet, but if we have one of these things, we're going to construct a Mach-modular form. So what we do is if we have one of these harmonic mass forms, we define a map called the shadow map by s of f hat is equal y to the k df hat by d tau bar, and then we take the complex conjugate of that quantity, a k-form. All right, so now let's notice that if f hat happens to be a weight k-modular form, weakly holomorphic, then s on this gives 0, because if f hat is weakly holomorphic, then d by d tau bar vanishes, and so the shadow map kills it, and if as examples related to moonshine that I'll discuss, affect the image of the shadow map vanishes as we go off to i infinity, then you can invert the shadow map very explicitly. So in other words, if you have some oh, I forgot to mention something over here, it follows from the fact that f hat obeys the it's annihilated by the weight k-laplacian and this formula that s of f hat is actually a weight 2 minus k-modular form. So the fact that this is holomorphic follows from the fact that it's annihilated by this weight k-modular form because this part of it is basically just the shadow map up to a complex conjugation. And using the transformation properties of this derivative in y to the k, you can check that this has weight 2 minus k if f hat has weight k. So this produces a weight 2 minus k-modular form and if you have one of these harmonic mass forms, then you can invert the shadow map by doing an integral. There's a proportionality constant here which I won't worry about. And there's some annoying business here which has to do with the fact that you complex conjugate here. You don't, you could reformulate everything without doing this, but then you would get modular forms that were anti-holomorphic, which are not sort of as natural. I mean, they're perfectly natural, but they're not what mathematicians are used to. There's some conjugation. There's some funny business here where you have to complex conjugate the argument, put a minus sign, and then complex conjugate the whole thing. But if you simply act on this integral with the shadow map, you will see that you get g back if you choose the proportionality constant correctly. And so this inverts the map. That is if you have a g star here, the shadow map gives you g. You can invert this to extract the terms of g. Weekly holomorphic means you're allowed to have a finite number of terms that have negative powers of q. All right, so it takes a few lines of computation, but you can check using the statements that I've given you that you now have the following nice situation. If you have one of these weak, harmonic mass forms and the shadow map gives you a g which is weight 2 minus k modular form, then the difference between f hat and g star is holomorphic because if I take d by d tau bar on it up to a factor of y to the k, that's just the shadow map and from this, these two terms cancel out, so this is holomorphic. f hat is f plus g star and this is modular meaning it transforms like a weight k modular form, but it is not holomorphic unless g is equal to zero. f is holomorphic but it's not modular. So this is a situation that's kind of like the holomorphic anomaly in supersymmetric gauge theories. There's a tension between two things you would like to be true. We're used to dealing with holomorphic functions and that often arises as a condition of supersymmetry or in modular forms because in the elliptic genus we've canceled out the q bar dependence by having some ground state condition and you would like things to be modular but in this situation you can't have both. You either have something that's holomorphic and not modular or modular but not holomorphic whenever this shadow g is non-zero. Yeah, I mean this is an inverse but it's not a unique inverse. No, it's two minus k. Okay, so the pair f and g is called a Mach modular form and it's shadow and f hat is called the completion of f. So in this description f is sort of the holomorphic part of a harmonic mass form. So this gives you kind of an abstract definition and you could then say well, how do you actually construct these things what are examples of them? Well so the first example is that a modular form is a Mach modular form with zero shadow. So okay, good. So it's a generalization of the notion of a modular form if the shadow is zero then there's no problem it's, you know, you have something that's already holomorphic. Now this is not entirely trivial so for example J of tau that showed up in Monstrous Moonshine is a weight zero modular form. You might ask yourself why couldn't we have had a Mach modular form? I mean could we, could it be some kind of Moonshine where there's a Mach modular form? Well because the upper half plane mod SL2z is genus zero it tells us that there are no weight two modular forms. If there were weight two modular forms they could be the shadow of something that's weight zero and there could be weight zero Mach modular forms but because the space of weight two modular forms vanishes J has to be modular. So this kind of reasoning can be extended in other situations where the genus zero property is kind of connected to modularity or Mach modularity when the space of some forms that could be shadows vanishes then you are sort of forced to land on modular forms. Now a second example which is slightly less trivial but not too much less trivial is to look at Eisenstein series. So it's a famous fact that when two K is even and K is greater than one this defines a weight two K modular form and on the other hand G2 is not convergent or not absolutely convergent showing that it's a modular form involves rearranging terms in the sum which you're not allowed to do if it's not absolutely convergent. So it's a fairly famous fact that you can you can choose a particular way of ordering this series where you now define G2 the sum over N squared plus the sum M not equal to zero sum on N on over M tau plus N squared and this quantity G2 hat transforms like a weight two modular form but it's not holomorphic because of this term involving the one over M tau and so G2 hat is a mock modular form with constant shadow. So that's the definition and it would now be nice to discuss in detail how to construct other examples but I'm not going to do that I'm just going to tell you that you can show that this quantity H2 of tau which exhibits this connection to M24 which is still somewhat mysterious is a weight one half mock modular form 24 times 8 cubed of tau notice that this is weight three halves so this is indeed an example where this guy is weight a half this is weight 2 minus K, 2 minus a half is three halves and you can also show that when you look at this Mackay Thompson series the twisted versions are also mock modular forms and you do that by showing that for each G in M24 and with Kai of G being the character of G in well M24 has a 23 dimensional representation and a one dimensional representation so a four dimensional reducible representation let Kai of G be the character of G in that representation then you can show that these twisted versions are given by Kai of G over 24 times H2 of tau minus TG of tau where TG of tau is a weight two modular form for gamma naught of N and in the simplest cases N is the order of G in slightly more complicated cases it's something a little more complicated than that but this part is modular this part is mock modular because H2 is mock modular unless Kai of G is equal to zero and Kai of G is equal to zero for some classes and then H2 of G is actually modular and not mock modular so the papers that I described earlier who looked at the twined versions of this showed that there is a decomposition of these terms into representations which preserves the mock modularity in this sense that the twined guys are still mock modular forms when Kai of G is non-zero they're modular when Kai of G is zero by explicitly constructing these TG of tau for all conjugacy classes of the monster that's the question that you asked earlier yeah there is a characterization in terms of in terms of I think in terms of their decomposition with respect to M23 or something like that I forget exactly what the statement is alright so the moon shine story here is more obscure because we don't have an explicit conformal field theory exhibiting the symmetry it's mathematically more complicated because it involves these mock modular forms rather than modular forms and while you can manipulate them sort of like modular forms now they're a little more complicated to deal with so the question is can we somehow use this mathematical formulation or something involving K3 to understand what's going on and in trying to do that it's natural to either think like a physicist or to think like a mathematician no there is no genus zero these guys are not weight zero they're weight two so no if you set G equal identity this vanishes and Kai of G is 24 because it's a 24 dimensional representation you're just looking at the character of the identity and you just recover H2 there is a genus there is a genus zero property here but to describe it I think I need to discuss the extension to umbral moonshine a little more visible in that broader context so well maybe by a physicist here what I really mean is string theorist yeah not really because when you look at this when you look at H2 alright I'm just making something up I didn't actually remember that these guys you look at them and you go alright there's a 45 dimensional irrep there's a 231 there's a 770 it's sort of obvious what to do but when you get to something like this it's not obvious what to do there are a finite number of irreducible representations and you can probably write this number as a sum of irreducible representations with positive coefficients in you know 35 different ways which way are you supposed to write it you don't know and each way of writing it will give you a different twined guy so you can't predict in advance what you're going to get all you can do is you can look at the first so many terms you can try to match it with something that has mock modular properties and then you can use that to predict what the higher order term should be in terms of the higher order terms in that Q expansion no in general you won't get something that's modular at all I mean suppose I looked at J of tau and instead of 1, 9, 6, 8, 8, 3 you know being that representation of the monster it was 1, 9, 6, 8, 8, 3 times the identity but I used some other decomposition for higher guys you know when you twined it you would just get something that was not modular at all but it didn't come from a torus partition function it came from taking the elliptic genus and doing this weird decomposition throwing away the massless characters keeping them massive well I agree if you can identify a Hilbert space of states that these numbers are counting positive multiplicities and that is connected with the conformal field there then yes but we don't have that but it's I don't I mean the partition function should be weight zero this is way to half and it's not even modular it's mock modular so I really don't know how to answer your question what why isn't what an inconsistency I don't understand go ahead yes yeah if I had such a situation so exactly so if I had a conformal field theory and I'm computing an insertion of G this looks like this now we know that when we take if we do modular transformations this goes to some G to some power G to some power and I can ask what transformations leave that fixed and I believe the answer is that if you look at gammonaut of N where N is the order of G then G1 goes to G1 and so that this should be a modular form for gammonaut of N if we had a conformal field theory and we had the partition function of it but we don't we don't so we can't conclude I mean it's not well yeah it's not modular it's Mach modular so we I mean it's it's true it's I mean it's even so it's it's true that there are these H2 of G's and even worse there are H2's of G's and H where H and G are commuting pairs of elements in the in M24 that you can define just like you'd expect to be able to define twisted and twine guys in a conformal field theory but the construction of these is all purely through manipulating Mach modular forms and not through CFT constructions which is why it suggests that there's some CFT way of understanding it but nobody knows what that is yet so I mean I'm sympathetic to the desire you're expressing but I don't so all right so I still have half an hour right is that right what time does it 25 minutes well I can't possibly say that much but if you I think the most natural thing to think is since we found evidence for M24 in the elliptic genus of K3 we should look at the elliptic genus for X being some other Kalabiow manifolds and it might actually be more natural to look at you know two end folds so things that have some hypercaler structure so that we have an n equals 4 but as far as I know that doesn't seem to lead to interesting generalizations where you find evidence for some other kind of group but if you think more just mathematically you could ask can we generalize quantity 501 which appears in the elliptic genus of K3 and not worry whether this generalization is connected to the elliptic genus of something but just whether mathematically it gives something that generalizes this construction and the answer is yes and so this is work done with Randa Chang and John Duncan and fired by and following some results by Atish, Samir, Murthy and Don Zaghe you can find the following structure which was found somewhat experimentally and then once we sort of recognize the pattern you can see how everything fit together so for each X which is the root lattice of a Neymar lattice I discussed in the first lecture so there are 23 of these for each one you can construct a whole bevy of mathematical objects so first of all you can construct LX which is the Neymar lattice to do this you have to add in various weights in the weight lattice corresponding to the root lattice but there's a way of doing that you can construct a pair HX and SX which are vector valued Mach modular form and it's shadow with M of X minus 1 components where M of X is the coxsweter number of X these exhibit moonshine in the same way that H2 did for M24 for a group DX which is the automorphism group of the Neymar lattice divided by the vial group of X generated by reflections in the roots and for X equals 1 to the 24th DX is M24 for X equals a 2 to the 12th which is another one of the root systems for Neymar lattice DX has a normal subgroup and quotient by that gives another Matthew group M12 if X is equal to E8 cubed DX is just the permutation group on three objects because you just permute the three copies of E8 so you get sporadic groups extensions of sporadic groups sometimes you get rather small groups in this pattern and you can also construct Gamma X and TX which is a genus zero subgroup of SL2Z and it's help module so here the genus zero characterization is not so much in these Mackay Thompson series but it's like each example that has a mock module or form and a group exhibiting moonshine for it is associated to a particular genus zero subgroup and I haven't told you how to construct these yet but they can be constructed from essentially the eigenvalues of the coxeter element in terms of a ratio of eta functions now I want to say just a few I guess what do I want to say it doesn't particularly it's just you can do it so I want to say a couple more things about the nature of this construction then I want to make some general comments so to construct HX use the same idea that we did in the elliptic genus but you have to choose certain special weight zero and index m-1 Jacobi forms into n equals 4 characters and these n equals 4 characters are labeled as before but now L goes from a half up to m-1 over 2 so you get vector valued guys where r can go from 1 up to m-1 and I'll just give you a couple of examples of what these Jacobi forms are if fi equal to theta i of tau and z over theta i of tau for i equals 2, 3 and 4 then we saw that the elliptic genus of k3 after being corrected for a stupid mistake was f2 squared plus f3 squared plus f4 squared it turns out that this is the guy associated a1 to the 24 for a2 to the 12th the corresponding form has weight zero and index 2 and is given by 4 f2 squared f3 squared plus f3 squared f4 squared plus f4 squared f2 squared and for a3 to the 8th this and these all are chosen such that there is a condition that tells you what kind of negative powers of q are allowed and they tell you that q to the 1 over 4m times hrx is always order 1 so you're never allowed to have anything more singular than q to the minus 1 over 4m and that strongly restricts the forms that are allowed here and it can be viewed as really a growth condition on the coefficients of the Jacobi form a growth condition which was used in the paper. So there are a lot of details here but I want to try to say something before I have to quit about what the general issues are and how people are trying to address them. So one of the main problems in trying to understand what's going on is that the of the mock modular forms hx or the Jacobi forms that they're built from is not really connected the construction of the group but somehow nonetheless you can check that they exhibit moonshine for gx by explicitly constructing the twined guise and verifying the mock modularity so in other words this group is determined by a Nehmeier lattice the coefficients of these mock modular forms have nice decompositions into representations of that group you can twine you can get things that are mock modular but it's not clear why this group is acting it's just completely obscure. Well it's natural to try to find a connection between them so one of the central problems is how do we connect the root system or the Nehmeier lattice lx to there's a mysterious connection so if you try to think about this from the point of view of conformal field theory you ask yourself well it has an ADE classification it's given by ADE components with total rank 24 and equal coaxial number h of x is a mock modular form is there some framework that we know in conformal field theory that has an ADE classification and that involves mock modular forms and mock modular forms occur things closely related to them occur as the elliptic genus in non-compact conformal field theories the most famous example which has been well studied is the cigar conformal field theory where you take sl2r mod u1 and the elliptic genus of this theory was investigated by troust and Ashok and Eguchi and Sugawara and what happens in a non-compact theory like this is you have the usual discrete states of a compact conformal field theory but then at some point a continuum of scattering states comes in just like in quantum mechanics problems where you have both bound states and scattering states and for the discrete states you will get a contribution to the elliptic genus which is holomorphic by the usual argument but it's not modular because it doesn't include the full spectrum of the theory and if you don't sum over the full spectrum you don't expect to get something modular if you compute the elliptic genus with the full spectrum you get an z-elliptic which is modular but it's not holomorphic because in a supersymmetric theory while supersymmetry pairs energies bosons and fermions with the same energy it doesn't require that you have the same spectral densities of fermions and bosons and that difference in the spectral density when you integrate over the continuum ends up giving you a contribution to the elliptic genus which is not holomorphic so you have exactly the same tension that occurs in mock modular forms and so it's natural to look for a non-compact conformal field theory that has some kind of ADE classification and we know examples of that they have dual descriptions and you can either think of a theory where you take five brains which have an ADE classification which is the ADE classification of the 2-0 theory and you can wrap them on K3 or you could look at C2 mod G for G a discrete subgroup of SU2 which is a local model for the kind of singularities that you can have in a K3 surface which have an ADE classification and you could use the macaic correspondence between cyclic groups, dihedral subgroups the orthogonal tetrahedral group octahedral group and the icosatedral group and AN DN or D2N E6, E7 and E8 and these kinds of theories have investigated by Samir Murthy and I Mezoroglu Chang and Harrison and recently by Agucci and Sugawara and you get results that are suggestive but so far not sort of on the nose you can get things that have maca mod or properties you can get various groups acting but you don't so far get the umbral groups on the nose acting on the maca mod or forms of umbral moonshine so it may be that something along these lines works and it just has to be more refined or it could be that this is sort of barking up the wrong tree and there's some other modification um there also seem to be connections to three-dimensional gravity which I'm not I don't think have time to explore or to discuss there's a really interesting unsolved problem there's a rich mathematical structure which generalizes the connection between k3 and m24 and involves maca mod or forms, special groups nemir lattices, things that often occur in string theory but no one has figured out how to put all the ingredients together to really explain what's going on and I think if we figure out how to do that we'll probably learn something interesting about string theory maybe some new kind of construction in the same way that monstrous moonshine really used an asymmetric orbifold construction before physicists knew what asymmetric orbifolds were and if we'd really understood that deeply we would have seen that there were interesting generalizations so my hope is that we'll learn something interesting about 3D gravity or black holes or new constructions in formal field theory if we really figure out what's going on here and I'm sorry that I haven't had the time to go into complete detail but I've probably been more technical than I should have been already so I think I'll just end here and ask if there are questions